
\chapter{Interest Rate Derivatives: The Black's Model}

Interest rate derivatives are instruments whose payoffs are dependent in some way on the level of
interest rates. Interest rate derivatives are more difficult to value than equity and foreign
exchange derivatives. There are a number of reasons for this:

\begin{my_itemize}
  \item The behavior of an individual interest rate is more complicated than that of a stock price
        or an exchange rate.
  \item For the valuation of many products, it is necessary to develop a model describing the
        behavior of the entire zero-coupon yield curve.
  \item The volatilities of different points on the yield curve are different.
  \item Interest rates are used for discounting as well as for defining the payoff from the
        derivative.
\end{my_itemize}

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\section{Black's Model}

Consider a European call option on a variable whose value is $V$. Define:

\begin{my_itemize}
  \item $T$: Time to maturity of the option.
  \item $F$: Forward price of $V$ for a contract with maturity $T$.
  \item $F_0$: Value of $F$ at time zero.
  \item $K$: Strike price of the option.
  \item $P(t, T)$: Price at time $t$ of a zero-coupon bound paying \$1 at time $T$.
  \item $V_T$: Value of $V$ at time $T$.
  \item $\sigma$: Volatility of $F$.
\end{my_itemize}

Black's model calculates the expected payoff from the option assuming:

\begin{my_itemize}
  \item $V_T$ has a lognormal distribution with the standard deviation of $ln V_T$ equal to
        $\sigma \sqrt{T}$: $ln V_T \sim \phi (E(ln V_T), \sigma \sqrt{T})$
  \item The expected value of $V_T$ is $F_0$: $E(V_T) = F_0$
\end{my_itemize}

It then discounts the expected payoff at the $T$-year risk-free rate by multiplying by $P(0, T)$.
The payoff from the option is $max(V_T - K, 0)$ at time $T$. Using the lemma \eqref{lemma_for_bs},
we have:

\[ E \Big( max(V_T - K, 0) \Big) = F_0 N(d_1) - K N(d_2) \]

So the value of the European call option is:

\begin{equation}
  c = P(0, T) \Big( F_0 N(d_1) - K N(d_2) \Big)
  \label{black_call}
\end{equation}

where:

\[ d_1 = \frac{ ln(F_0/K) + \sigma^2 T / 2 }{ \sigma \sqrt{T} } \]

\[ d_2 = \frac{ ln(F_0/K) - \sigma^2 T / 2 }{ \sigma \sqrt{T} } = d_1 - \sigma \sqrt{T} \]


Similarly, the value $p$ of the corresponding put option is given by:

\begin{equation}
  p = P(0, T) \Big( K N(-d_2) - F_0 N(-d_1) \Big)
  \label{black_put}
\end{equation}

We can extend Black's model to allow for the situation where the payoff is calculated from the value
of the variable $V$ at time $T$, but the payoff is actually made at some later time $T^*$. The
expected payoff is discounted from time $T^*$ instead of time $T$, so that:

\begin{equation}
  c = P(0, T^*) \Big( F_0 N(d_1) - K N(d_2) \Big)
  \label{black_call_2}
\end{equation}

\begin{equation}
  p = P(0, T^*) \Big( K N(-d_2) - F_0 N(-d_1) \Big)
  \label{black_put_2}
\end{equation}

where:

\[ d_1 = \frac{ ln(F_0/K) + \sigma^2 T / 2 }{ \sigma \sqrt{T} } \]

\[ d_2 = \frac{ ln(F_0/K) - \sigma^2 T / 2 }{ \sigma \sqrt{T} } = d_1 - \sigma \sqrt{T} \]

An important feature of Black's model is that we do not have to assume geometric Brownian motion for
the evolution of either $V$ or $F$. All that we require is that $V_T$ be lognormal at time $T$. The
parameter $\sigma$ is usually referred to as the volatility of $F$ or the forward volatility of $V$.
However, its only role is to define the standard deviation of $ln V_T$ by means of the relationship:

\[ Standard~deviation~of~ln~V_T = \sigma \sqrt{T} \]

The volatility parameter does not necessarily say anything about the standard deviation of $ln V$ at
times other than time $T$.

\textcolor{red}{validity of Black's Model: see p510.}

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\section{Bond Options}

A bond option is an option to buy or sell a particular bond by a certain date for a particular
price.

\textcolor{red}{more about bond option: see p511.}

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\section{Interest Rate Caps}

Consider a floating rate note where the interest rate is periodically reset equal to LIBOR. The time
between resets is known as the tenor. An interest rate cap is designed to provide insurance against
the rate of interest on the floating rate not rising above a certain level. This level is known as
the cap rate.

Consider a cap with a total life of $T$, a principal of $L$, and a cap rate of $R_K$. Suppose that
the reset dates are $t_1, t_2, ..., t_n$ and define $t_{n+1} = T$. Define $R_k$ as the interest rate
for the period between time $t_k$ and $t_{k+1}$ observed at time $t_k$ ($1 \leq k \leq n$). The cap
leads to a payoff at time $t_{k+1}$ ($k = 1, 2, ..., n$) of:

\begin{equation}
  L \delta_k max(R_k - R_K, 0)
  \label{payoff_caplet}
\end{equation}

where $\delta_k = t_{k+1} - t_k$. (Both $R_k$ and $R_K$ are expressed with a compounding frequency
equal to the frequency of resets.)

Equation \eqref{payoff_caplet} is a call option on the LIBOR rate observed at time $t_k$ with the
payoff occurring at time $t_{k+1}$. The cap is a portfolio of $n$ such options. LIBOR rates are
observed at times $t_1, t_2, ..., t_{n+1}$. The $n$ call options underlying the cap are known as
caplets.

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\section{Interest Rate Floors and Collars}

\textcolor{red}{See p517.}

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\section{Valuation of Caps and Floors}

As shown in equation \eqref{payoff_caplet}, the caplet corresponding to the rate observed at time
$t_k$ provides a payoff at time $t_{k+1}$ of:

\[ L \delta_k max(R_k - R_K, 0) \]

If the rate $R_k$ is assumed to be lognormal with volatility $\sigma_k$, equation
\eqref{black_call_2} gives the value of this caplet as:

\begin{equation}
  L \delta_k P(0, t_{k+1}) \Big( F_k N(d_1) - R_K N(d_2) \Big)
  \label{black_caplet}
\end{equation}

where:

\[ d_1 = \frac{ ln(F_k/R_K) + \sigma_k^2 t_k / 2 }{ \sigma_k \sqrt{t_k} } \]
\[ d_2 = \frac{ ln(F_k/R_K) - \sigma_k^2 t_k / 2 }{ \sigma_k \sqrt{t_k} }
       = d_1 - \sigma_k \sqrt{t_k} \]

and $F_k$ is the forward rate for the period between time $t_k$ and $t_{k+1}$. The value of the
corresponding floorlet is, from equation \eqref{black_put_2}:

\begin{equation}
  L \delta_k P(0, t_{k+1}) \Big( R_K N(-d_2) - F_k N(-d_1) \Big)
  \label{black_floorlet}
\end{equation}

Note that $R_K$ and $F_k$ are expressed with a compounding frequency equal to the frequency of
resets in these equations.

Each caplet of a cap must be valued separately using equation \eqref{black_caplet}. One approach is
to use a different volatility for each caplet. The volatilities are then referred as spot
volatilities. An alternative approach is to use the same volatility for all the caplets constituting
any particular cap but to vary this volatility according to the life of the cap. The volatilities
used are then referred to as flat volatilities. The volatilities quoted in the market are usually
flat volatilities.


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